# Phonon

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In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. Often designated a quasiparticle, [1] it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

In physics, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate forces are applied to them. If the material is elastic, the object will return to its initial shape and size when these forces are removed. Hooke's law states that the force should be proportional to the extension. The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied. When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.

An atom is the smallest constituent unit of ordinary matter that constitutes a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms. Atoms are extremely small; typical sizes are around 100 picometers. They are so small that accurately predicting their behavior using classical physics – as if they were billiard balls, for example – is not possible. This is due to quantum effects. Current atomic models now use quantum principles to better explain and predict this behavior.

## Contents

Phonons play a major role in many of the physical properties of condensed matter, such as thermal conductivity and electrical conductivity. The study of phonons is an important part of condensed matter physics.

The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by , , or .

The concept of phonons was introduced in 1932 by Soviet physicist Igor Tamm. The name phonon comes from the Greek word φωνή (phonē), which translates to sound or voice because long-wavelength phonons give rise to sound. The name is analogous to the word photon .

The Soviet Union, officially the Union of Soviet Socialist Republics (USSR), was a Marxist-Leninist sovereign state in Eurasia that existed from 1922 to 1991. Nominally a union of multiple national Soviet republics, its government and economy were highly centralized. The country was a one-party state, governed by the Communist Party with Moscow as its capital in its largest republic, the Russian Soviet Federative Socialist Republic. Other major urban centres were Leningrad, Kiev, Minsk, Tashkent, Alma-Ata, and Novosibirsk. It spanned over 10,000 kilometres (6,200 mi) east to west across 11 time zones, and over 7,200 kilometres (4,500 mi) north to south. It had five climate zones: tundra, taiga, steppes, desert and mountains.

Igor Yevgenyevich Tamm was a Soviet physicist who received the 1958 Nobel Prize in Physics, jointly with Pavel Alekseyevich Cherenkov and Ilya Mikhailovich Frank, for their 1934 discovery of Cherenkov radiation.

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

## Definition

A phonon is the quantum mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency. [2] In classical mechanics this designates a normal mode of vibration. Normal modes are important because any arbitrary lattice vibration can be considered to be a superposition of these elementary vibration modes (cf. Fourier analysis). While normal modes are wave-like phenomena in classical mechanics, phonons have particle-like properties too, in a way related to the wave–particle duality of quantum mechanics.

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. Examples of lattice models in condensed matter physics include the Ising model, the Potts model, the XY model, the Toda lattice. The exact solution to many of these models includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang-Baxter equation and quantum groups. The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations. An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics. However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers. Examples include the bond fluctuation model and the 2nd model.

## Lattice dynamics

The equations in this section do not use axioms of quantum mechanics but instead use relations for which there exists a direct correspondence in classical mechanics.

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.

For example: a rigid regular, crystalline (not amorphous), lattice is composed of N particles. These particles may be atoms or molecules. N is a large number, say of the order of 1023, or on the order of Avogadro's number for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position. These forces may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric force. Magnetic and gravitational forces are generally negligible. The forces between each pair of atoms may be characterized by a potential energy function V that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting: [3]

In condensed matter physics and materials science, an amorphous or non-crystalline solid is a solid that lacks the long-range order that is characteristic of a crystal. In some older books, the term has been used synonymously with glass. Nowadays, "glassy solid" or "amorphous solid" is considered to be the overarching concept, and glass the more special case: Glass is an amorphous solid that exhibits a glass transition. Polymers are often amorphous. Other types of amorphous solids include gels, thin films, and nanostructured materials such as glass.

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

In molecular physics, the van der Waals force, named after Dutch scientist Johannes Diderik van der Waals, is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and therefore more susceptible to disturbance. The van der Waals force quickly vanishes at longer distances between interacting molecules.

${\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)}$

where ri is the position of the ith atom, and V is the potential energy between two atoms.

It is difficult to solve this many-body problem explicitly in either classical or quantum mechanics. In order to simplify the task, two important approximations are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively screened. Secondly, the potentials V are treated as harmonic potentials. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by Taylor expanding V about its equilibrium value to quadratic order, giving V proportional to the displacement x2 and the elastic force simply proportional to x. The error in ignoring higher order terms remains small if x remains close to the equilibrium position.

The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling phonons. (For other common lattices, see crystal structure.)

The potential energy of the lattice may now be written as

${\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.}$

Here, ω is the natural frequency of the harmonic potentials, which are assumed to be the same since the lattice is regular. Ri is the position coordinate of the ith atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).

### Lattice waves

Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength λ is marked.

There is a minimum possible wavelength, given by twice the equilibrium separation a between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2a, due to the periodicity of the lattice. This can be thought as one consequence of Nyquist–Shannon sampling theorem, the lattice points are viewed as the "sampling points" of a continuous wave.

Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes do possess well-defined wavelengths and frequencies.

### One-dimensional lattice

In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.

#### Classical treatment

The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step (adiabatic approximation):

n − 1  n n + 1   a

···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···

→→  →→→
un − 1 un un + 1

where n labels the nth atom out of a total of N, a is the distance between atoms when the chain is in equilibrium, and un the displacement of the nth atom from its equilibrium position.

If C is the elastic constant of the spring and m the mass of the atom, then the equation of motion of the nth atom is

${\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}$

This is a set of coupled equations. Since the solutions are expected to be oscillatory, new coordinates are defined by a discrete Fourier transform, in order to decouple them. [4]

Put

${\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}}$

Here, na corresponds and devolves to the continuous variable x of scalar field theory. The Qk are known as the normal coordinates, continuum field modes φk. Substitution into the equation of motion produces the following decoupled equations (this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform [5] ,

${\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}$

These are the equations for harmonic oscillators which have the solution

${\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}}$

Each normal coordinate Qk represents an independent vibrational mode of the lattice with wavenumber k which is known as a normal mode.

The second equation, for ωk, is known as the dispersion relation between the angular frequency and the wavenumber. In the continuum limit, a→0, N→∞, with Na held fixed, unφ(x), a scalar field, and ${\displaystyle \omega (k)\propto ka}$. This amounts to free scalar classical field theory.

#### Quantum treatment

A one-dimensional quantum mechanical harmonic chain consists of N identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.

In some contrast to the previous section, the positions of the masses are not denoted by ui, but, instead, by x1, x2…, as measured from their equilibrium positions (i.e. xi = 0 if particle i is at its equilibrium position.) In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is

${\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}$

where m is the mass of each atom (assuming it is equal for all), and xi and pi are the position and momentum operators, respectively, for the ith atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with waves in Fourier space which uses normal modes of the wavevector as variables instead coordinates of particles. The number of normal modes is same as the number of particles. However, the Fourier space is very useful given the periodicity of the system.

A set of N "normal coordinates" Qk may be introduced, defined as the discrete Fourier transforms of the xk and N "conjugate momenta" Πk defined as the Fourier transforms of the pk:

{\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}

The quantity kn turns out to be the wavenumber of the phonon, i.e. 2π divided by the wavelength.

This choice retains the desired commutation relations in either real space or wavevector space

{\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left[x_{l},p_{m}\right]\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0\end{aligned}}}

From the general result

{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}

The potential energy term is

${\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}$

where

${\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}$

The Hamiltonian may be written in wavevector space as

${\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}$

The couplings between the position variables have been transformed away; if the Q and Π were Hermitian (which they are not), the transformed Hamiltonian would describe N uncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, periodic boundary conditions are imposed, defining the (N + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

${\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }$

The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode ωk are:

${\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }$

The levels are evenly spaced at:

${\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }$

where 1/2ħω is the zero-point energy of a quantum harmonic oscillator.

An exact amount of energy ħω must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field is quantized, the quantum of vibrational energy is called a phonon.

All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later. [6]

### Three-dimensional lattice

This may be generalized to a three-dimensional lattice. The wavenumber k is replaced by a three-dimensional wavevector k. Furthermore, each k is now associated with three normal coordinates.

The new indices s = 1, 2, 3 label the polarization of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

### Dispersion relation

For a one-dimensional alternating array of two types of ion or atom of mass m1, m2 repeated periodically at a distance a, connected by springs of spring constant K, two modes of vibration result: [8]

${\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}$

where k is the wavevector of the vibration related to its wavelength by ${\displaystyle k={\tfrac {2\pi }{\lambda }}}$.

The connection between frequency and wavevector, ω = ω(k), is known as a dispersion relation. The plus sign results in the so-called optical mode, and the minus sign to the acoustic mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

The speed of propagation of an acoustic phonon, which is also the speed of sound in the lattice, is given by the slope of the acoustic dispersion relation, ωk/k (see group velocity.) At low values of k (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ωa, independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of k, i.e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in its primitive cell, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wavevector. The boundaries at −π/a and π/a are those of the first Brillouin zone. [8] A crystal with N  2 different atoms in the primitive cell exhibits three acoustic modes: one longitudinal acoustic mode and two transverse acoustic modes. The number of optical modes is 3N  3. The lower figure shows the dispersion relations for several phonon modes in GaAs as a function of wavevector k in the principal directions of its Brillouin zone. [7]

Many phonon dispersion curves have been measured by inelastic neutron scattering.

The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support shear stresses (but see viscoelastic fluids, which only apply to high frequencies).

### Interpretation of phonons using second quantization techniques

In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an operator, then it describes a quantum field theory of non-interacting bosons.

The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem. We introduce a set of ladder operators defined by:[ citation needed ]

{\displaystyle {\begin{alignedat}{2}b_{k}&={\frac {1}{\sqrt {2}}}\left({\frac {Q_{k}}{l_{k}}}+i{\frac {\Pi _{-k}}{\frac {\hbar }{l_{k}}}}\right),&\quad Q_{k}&=l_{k}{\frac {1}{\sqrt {2}}}\left({b_{k}}^{\dagger }+b_{-k}\right)\\{b_{k}}^{\dagger }&={\frac {1}{\sqrt {2}}}\left({\frac {Q_{-k}}{l_{k}}}-i{\frac {\Pi _{k}}{\frac {\hbar }{l_{k}}}}\right),&\quad \Pi _{k}&={\frac {\hbar }{l_{k}}}{\frac {i}{\sqrt {2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)\\l_{k}&={\sqrt {\frac {\hbar }{m\omega _{k}}}}\end{alignedat}}}

By direct insertion on the Hamiltonian, it is readily verified that[ citation needed ]

${\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right),\quad \left[b_{k},{b_{k'}}^{\dagger }\right]=\delta _{k,k'},\quad {\Big [}b_{k},b_{k'}{\Big ]}=\left[{b_{k}}^{\dagger },{b_{k'}}^{\dagger }\right]=0.}$

As with the quantum harmonic oscillator, one can show that bk and bk respectively create and destroy one excitation of energy ħωk. These excitations are phonons.[ citation needed ]

Two important properties of phonons may be deduced. Firstly, phonons are bosons, since any number of identical excitations can be created by repeated application of the creation operator bk. Secondly, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the ladder operators contain sums over the position and momentum operators of every atom.[ citation needed ]

It is not a priori obvious that these excitations generated by the b operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the position–position correlation function .[ citation needed ] Let |k denote a state with a single quantum of mode k excited, i.e.

${\displaystyle |k\rangle =b_{k}^{\dagger }|0\rangle .}$

One can show that, for any two atoms j and l,

${\displaystyle \langle k|x_{j}(t)x_{l}(0)|k\rangle ={\frac {\hbar }{Nm\omega _{k}}}\cos {\big (}k(j-l)a-\omega _{k}t{\big )}+\langle 0|x_{j}(t)x_{l}(0)|0\rangle }$

which has the form of a lattice wave with frequency ωk and wavenumber k.

In three dimensions the Hamiltonian has the form

${\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}$[ citation needed ]

## Acoustic and optical phonons

Solids with more than one atom in the smallest unit cell exhibit two types of phonons: acoustic phonons and optical phonons.

Acoustic phonons are coherent movements of atoms of the lattice out of their equilibrium positions. If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the propagation direction is comparable to waves on a string. If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero deformation energy. Acoustic phonons exhibit a linear relationship between frequency and phonon wavevector for long wavelengths. The frequencies of acoustic phonons tend to zero with longer wavelength. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.

Optical phonons are out-of-phase movements of the atoms in the lattice, one atom moving to the left, and its neighbour to the right. This occurs if the lattice basis consists of two or more atoms. They are called optical because in ionic crystals, like sodium chloride, they are excited by infrared radiation. The electric field of the light will move every positive sodium ion in the direction of the field, and every negative chloride ion in the other direction, sending the crystal vibrating.

Optical phonons have a non-zero frequency at the Brillouin zone center and show no dispersion near that long wavelength limit. This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. Optical phonons that interact in this way with light are called infrared active. Optical phonons that are Raman active can also interact indirectly with light, through Raman scattering. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse modes respectively; the splitting between LO and TO frequencies is often described accurately by the Lyddane–Sachs–Teller relation.

When measuring optical phonon energy by experiment, optical phonon frequencies are sometimes given in spectroscopic wavenumber notation, where the symbol ω represents ordinary frequency (not angular frequency), and is expressed in units of cm −1. The value is obtained by dividing the frequency by the speed of light in vacuum. In other words, the frequency in cm−1 units corresponds to the inverse of the wavelength of a photon in vacuum, that has the same frequency as the measured phonon. [9] The cm−1 is a unit of energy used frequently in the dispersion relations of both acoustic and optical phonons, see units of energy for more details and uses.

## Crystal momentum

By analogy to photons and matter waves, phonons have been treated with wavevector k as though it has a momentum ħk, [10] however, this is not strictly correct, because ħk is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because k is only determined up to addition of constant vectors (the reciprocal lattice vectors and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates Q and Π are defined so that

${\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}}$

where

${\displaystyle K={\frac {2n\pi }{a}}}$

for any integer n. A phonon with wavenumber k is thus equivalent to an infinite family of phonons with wavenumbers k ± 2π/a, k ± 4π/a, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. Bloch electrons obey a similar set of restrictions.

It is usually convenient to consider phonon wavevectors k which have the smallest magnitude |k| in their "family". The set of all such wavevectors defines the first Brillouin zone . Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.

## Thermodynamics

The thermodynamic properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the phonon dispersion relations combine in what is known as the phonon density of states which determines the heat capacity of a crystal. By the nature of this distribution, the heat capacity is dominated by the high-frequency part of the distribution, while thermal conductivity is primarily the result of the low-frequency region.[ citation needed ]

At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons. A lattice at a nonzero temperature has an energy that is not constant, but fluctuates randomly about some mean value. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. Because these phonons are generated by the temperature of the lattice, they are sometimes designated thermal phonons.[ citation needed ]

Thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. This behavior is an extension of the harmonic potential into the anharmonic regime. The behavior of thermal phonons is similar to the photon gas produced by an electromagnetic cavity, wherein photons may be emitted or absorbed by the cavity walls. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators, giving rise to Black-body radiation. Both gases obey the Bose–Einstein statistics: in thermal equilibrium and within the harmonic regime, the probability of finding phonons or photons in a given state with a given angular frequency is:[ citation needed ]

${\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}}$

where ωk,s is the frequency of the phonons (or photons) in the state, kB is Boltzmann's constant, and T is the temperature.

### Heat flow in nanometer-wide gaps

The idea of quantum tunneling applied to phonons produces the idea of phonon tunneling, where across nanometer-wide gaps, heat can transfer between materials from phonon that "tunnel" between the two materials. [11] The type of heat transfer works between distances too large for conduction to occur but too small for radiation to occur.

## Operator formalism

The phonon Hamiltonian is given by

${\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-{\tfrac {1}{2}}\hbar \omega _{\alpha }\right)}$

In terms of the operators, these are given by

${\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }}$

Here, in expressing the Hamiltonian in operator formalism, we have not taken into account the 1/2ħωq term, since if we take an infinite lattice or, for that matter a continuum, the 1/2ħωq terms will add up giving an infinity. Hence, it is "renormalized" by putting the factor of 1/2ħωq to 0 arguing that the difference in energy is what we measure and not the absolute value of it. Hence, the 1/2ħωq factor is absent in the operator formalised expression for the Hamiltonian.

The ground state also called the "vacuum state" is the state composed of no phonons. Hence, the energy of the ground state is 0. When, a system is in state |n1n2n3, we say there are nα phonons of type α. The nα are called the occupation number of the phonons. Energy of a single phonon of type α being ħωq, the total energy of a general phonon system is given by n1ħω1 + n2ħω2 +…. In other words, the phonons are non-interacting. The action of creation and annihilation operators are given by

${\displaystyle {a_{\alpha }}^{\dagger }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }+1}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }+1),n_{\alpha +1}\ldots {\Big \rangle }}$

and,

${\displaystyle a_{\alpha }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }-1),n_{\alpha +1},\ldots {\Big \rangle }}$

i.e. aα creates a phonon of type α while aα annihilates. Hence, they are respectively the creation and annihilation operator for phonons. Analogous to the quantum harmonic oscillator case, we can define particle number operator as

${\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.}$

The number operator commutes with a string of products of the creation and annihilation operators if, the number of a are equal to number of a.

Phonons are bosons, since |α,β = |β,α i.e. they are symmetric under exchange. [12]

## Nonlinearity

As well as photons, phonons can interact via parametric down conversion [13] and form squeezed coherent states. [14]

## Predicted properties

Even though phonons are often used as a quasiparticle, some popular research has shown that phonons and rotons may indeed have some kind of mass and be affected by gravity as standard particles are. [15] In particular, phonons are predicted to have a kind of negative mass and negative gravity. [16] This can be explained by how phonons are known to travel faster in denser materials. Because the part of a material pointing towards a gravitational source is closer to the object, it becomes denser on that end. From this, it is predicted that phonons would deflect away as it detects the difference in densities, exhibiting the qualities of a negative gravitational field. [17] Although the effect would be too small to measure, it is possible that future equipment could lead to successful results.

Phonons have also been predicted to play a key role in superconductivity in materials and the prediction of superconductive compounds. [18]

In 2019, researchers were able to isolate individual phonons without destroying them for the first time. [19]

## Related Research Articles

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was first proposed by Lev Landau in 1933 to describe an electron moving in a dielectric crystal where the atoms move from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the electron's effective mass.

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

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The vibrational partition function traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.

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In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters, between a state containing few electromagnetic excitations and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.

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